Applied Measure Theory for Probabilistic Modeling

TL;DR

This paper introduces MeasureTheory.jl, a Julia package that provides foundational measure-theoretic tools and concepts to enhance probabilistic programming and statistical computing, focusing on measures and densities.

Contribution

It develops a measure-centric framework with new notions like locally-dominated measures, improving the handling of densities and distributions in Julia.

Findings

Introduces a measure-based approach for probabilistic modeling.

Defines the concept of locally-dominated measures for better density management.

Enhances Julia's capabilities for probabilistic programming with new measure-theoretic tools.

Abstract

Probabilistic programming and statistical computing are vibrant areas in the development of the Julia programming language, but the underlying infrastructure dramatically predates recent developments. The goal of MeasureTheory.jl is to provide Julia with the right vocabulary and tools for these tasks. In the package we introduce a well-chosen set of notions from the foundations of probability together with powerful combinators and transforms, giving a gentle introduction to the concepts in this article. The task is foremost achieved by recognizing measure as the central object. This enables us to develop a proper concept of densities as objects relating measures with each others. As densities provide local perspective on measures, they are the key to efficient implementations. The need to preserve this computationally so important locality leads to the new notion of locally-dominated measure solving the so-called base measure problem and making work with densities and distributions in Julia easier and more flexible.